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Theorem eltrrels2 38287
Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
eltrrels2 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))

Proof of Theorem eltrrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dftrrels2 38283 . 2 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
2 coideq 37954 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
3 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
42, 3sseq12d 4012 . 2 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
51, 4rabeqel 37962 1 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  wss 3946  ccom 5676   Rels crels 37888   TrRels ctrrels 37900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-rels 38193  df-ssr 38206  df-trs 38280  df-trrels 38281
This theorem is referenced by:  eltrrelsrel  38289
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